Vivianni, and Stuff Like That

Vincenzo Viviani's birthday is coming up soon, and I was thinking about the common theorem named for him. Born on April 5, 1622, in 1639, at the age of 17, he became an assistant of Galileo Galilei in Arcetri. He remained a disciple until Galileo's death in 1642.  A note from Thony Christie informed me that after Galileo's death, his papers were being used by the local butcher to wrap his meat and sausages until Viviani rescued what was left of them.
From 1655 to 1656, Viviani edited the first edition of Galileo's collected works. He was a leader in his field and founded the Accademia del Cimento. As one of the first important scientific societies, this organization came before England's Royal Society.
The theorem that bears his name is a simple idea approachable to most good HS students. It says that from any point in the interior or on the periphery of an equilateral triangle, the sum of the perpendicular distances to the three sides is constant, and equal to the altitude.
The simple proof by dividing up the area of the interior should certainly be within the grasp of most good geometry students.
If the theorem is allowed to die there, it will probably have little impact on the student's development.  They'll lock it away in a curiosity file and little else will come of it.  But if they are allowed and encouraged to explore, the theorem may help them develop the habits of generalization and experimentation that assist their mathematical  development.
There are several nice related and similar ideas that occur around the theorem.  The first is the question of what happens when the point moves outside the triangle.

Certainly at this point the sum of the lengths no longer equal a constant.  But the curious student may begin to wonder how different is it, and why.  Perhaps a guided query from the teacher may be needed.  What was the key to the original proof? (please say area !)
How does that look now?

Why is the theorem about regular triangles.  Would it work with other regular n-gons?  How about a parallelogram? How about a rhombus? What about a trapezoid.  What about any quadrilateral in general?
What if we move the point outside in the quadrilateral as we did with the triangle?
What about a pentagon?  What about n-gons in general?

Do not overlook non-convex n-gons.  What about a dart shape.

Just a few weeks ago I came across a couple of other little interesting theorems which reach from the basics of HS geometry to more advanced math, but they are too similar in nature, and too beautiful not to include as an appetizer.

The first two are related and about a random point in a circle.  Construct a circle and pick a point at random inside it.  Now construct two perpendicular lines through the point and mark the intersections of the lines with the circle.  Can you prove that the two pairs (in red and blue) of opposite arcs sum to the same arc length?



It is less elementary to show that, in fact, the two opposite areas cut out by the perpendicular lines area also equal.  Both these properties are true if  2n lines in equal angles around the point cut the circle in 4n points.

And to go back to a look at Vivianni's theorem that may not seem at first glance to be at all the same, but can be shown by inverting Vivianni's Theorem.


If a regular triangle is inscribed in a circle, and a point on the circumference is picked between two Vertices and the distance from P to the vertices in order around the triangle are called d1, d2, and d3, then it will be true that  


In a similar way this can be extended to any regular n-gon.  I like the triangle one for the sort of "Pythagorean-like" quality of it, but it would probably be easier for younger students to confirm the truth with a square carefully placed on the coordinate grid.